RESEARC H ARTIC L E Open Access
Mechanistic insights from a quantitative analysis
of pollen tube guidance
Shannon F Stewman
1,2,9
, Matthew Jones-Rhoades
7
, Prabhakar Bhimalapuram
8
, Martin Tchernookov
2,6
,
Daphne Preuss
3,4,5
, Aaron R Dinner
1,2,5*
Abstract
Background: Plant biologists have long speculated about the mechanisms that guide pollen tubes to ovules.
Although there is now evidence that ovules emit a diffusible attractant, little is known about how this attractant
mediates interactions between the pollen tube and the ovules.
Results: We employ a semi-in vitro assay, in which ovules dissected from Arabidopsis thaliana are arranged around
a cut style on artificial medium, to elucidate how ovules release the attractant and how pollen tubes respond to it.
Analysis of microscopy images of the semi-in vitro system shows that pollen tubes are more attracted to ovules
that are incubated on the medium for longer times before pollen tubes emerge from the cut style. The responses
of tubes are consistent with their sensing a gradient of an attractant at 100-150 μm, farther than previously
reported. Our microscopy images also show that pollen tubes slow their growth near the micropyles of functional
ovules with a spatial range that depends on ovule incubation time.
Conclusions: We propose a stochastic model that captures these dynamics. In the model, a pollen tube senses a
difference in the fraction of receptors bound to an attractant and changes its direction of growth in response; the
attractant is continuously released from ovules and spreads isotropically on the medium. The model suggests that
the observed slowing greatly enhances the ability of pollen tubes to successfully target ovules. The relation of the

that fertilized ovules may emit a short-lived repulsive
signal to prevent multiple pollen tubes entering [7], and
nitric oxide has also been shown to repel pollen tubes
in in vitro [8] and semi-in vitro assays [9]. More
recently, it has been shown that the synergid cells of
Torenia fournieri secrete small peptides that induce
chemotropism [10]. Although these observations provide
* Correspondence:
1
Department of Chemistry, The University of Chicago, 929 E 57th St,
Chicago, IL 60637, USA
Stewman et al. BMC Plant Biology 2010, 10:32
/>© 2010 Stewman et al; licensee BioMed Central Ltd. This is an Open Acce ss article distributed under the terms of the Creative
Commons Attribution License (http://creativ ecommon s.org/licenses/by/2.0), which permits unrestricted use, distribution, and
reprodu ction in any medium, provided the origi nal work is properly cited.
evidence for diffusible attractants, the mechanisms of
action of t he participating molecules remain unknown,
as do their identities in most species. Furthermore, a
lack of detail in characterizing pollen tube responses has
complicated discussions of the range at which the gui-
dance operates and, in turn, the role of guidance in vivo.
A series of semi-in vitro experiments have provided
substantial evidence t hat diffusible signals that are
released by the ovule in vitro play a potentially impor-
tant role in later stages of guidance. In these experi-
ments, stigma are pollinated, cut, and placed on an agar
medium [7,10-13]. Ovules are dissected from the ovary
and a rranged around the cut end o f the stigma
(Fig. 1C). The pollen germinates on the stigma, grows
through the style, and emerges onto the surface of the

tubes often turned toward ovules, consistent with pollen
tubes following a gradient of an attractant by sensing a
change in the concentration of the attractant across
their tips.
To explo re the implications of these results, we devel-
oped a mathematical model of pollen tube response to a
gradient of a diffusible attractant that is continuously
released by the ovules. Bec ause little is known about the
receptors and internal signals that drive pollen tube
response to such attractants, our model makes no
assumptions about the molecular mechanism for sensing
this gradi ent and instead focuse s on whole-cell features,
an approach which has been used to model algae photo-
taxis [15], whole-cell motility [16,17], trajectories of
Listeria [18], and leukocyte chemotaxis [19-21]. The
model successfully captures both the directed and ran-
dom growth we observe experimentally and suggests
that the observed slowing of growth in vitro greatly
increases the ability of pollen tubes to target an ovule
succ essfully. The implica tions that our observations and
model have for guidance in vivo are discussed.
AB
st
st
ov
pt
pg
si
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f

Incubation time influences pollen tube response
Previous semi-in vitro work has shown that pollen tubes
approach the micropyle of functional ovules more fre-
quently than heat-treated ovules [11] or ovules with
laser-ablated cells [14]. More recent approaches have
quantified this apparent attraction by assessing how the
rate of in vitro fertilization changes when pollen tubes
are exposed to ovules dissected from closely-related spe-
cies[7,13].Herewepresentaquantitativeanalysisof
how pollen tubes grow and respond to dissected ovules
in vitro.
Dissected ovules from Arabidopsis thaliana plants
werearrangedaroundacutstyleusingaprocedure
adapted from [7] (see Methods). The cut styles were
pollinated such that between 20-40 pollen tubes even-
tually emerged from the style onto the medium, where
the tubes were then allowed to grow 30 minutes before
imaging was started (Table 1). Confocal stacks were
acquired every 20 m inutes for 320 minutes. To asse ss
pollen tube growth quantitatively, we tracked the posi-
tions of the pollen tube tips at each time point, and
used these positions to construct trajectories of tube
growth. These trajectories were combined with the loca-
tions of the micropyles of the ovules to give distance
and angle data, and data from stigmas w ith the same
incubation time were combined.
To assay the amount of attraction that pollen tubes had
toward an ovule, we calculated the fraction of pollen tube
tips that were within a certain distance of a micropyle
that grew either closer to (f

total
and f
farther
= N
farther
/N
total
.
Using this approach, we examined these frequencies for
ovules that had in cubated on the medium for 0, 2, and 4
hours. As a neg ative control, we used heat-treated ovules
that had been incubated for 2 hours. This incubation time
was chosen to be consistent with previous experiments in
Arabidopsis [7]. Palanivelu and Preuss had placed heat-
treated control ovules at the same time as pollinating the
cut style, which corresponds to an incubation time of 2
hours in our assays (Table 1). In each experiment, the cut
end of an ovary was placed a minimum of 2 50 μm (typi-
cally 380-430 μm) from the micropyle of an ovule; there
was no s ignficant difference (p > 0.1, o ne-way ANOVA)
between the average distances from the center of the cut
transmitting tract to each micropyle in any of the experi-
mental conditions (Table 1). We found that at all distances
(0-200 μm), the frequency with which tips moved farther
from a micropyle of an ovule decreased with the incuba-
tion time of that ovule (Fig. 2B, bottom). The trends were
very consistent: at all distances, the frequency of tips grow-
ing farther (f
farther
) from the micropyle of ovules that had

visible (Fig. 2B). This difference stems from the facts that
pollen tubes persist growing in the same direction for long
distances, and the direction of the cut style initially orients
the tubes to grow toward the ovules in the semi-in vitro
assay.
The previous statistics include pollen tube growth that
occurs both before and after the pollen tube penetrates
the ovule. The points after penetration were included to
allow an unbiased comparison with the heat-treated
control but may affect the trends in f
closer
and f
farther
.To
prevent polyspermy, the interactions between pollen
tubes and ovules change once an ovule is fertilized,
which occurs shortly after pollen tube penetration
[7,22-25]. We constructed frequencies
f
closer

and
f
farther

for f unctional ovules that only include points in
each pollen tube trajectory that and f
farther
correspond
to times before the nearest ovule was penetrated. The


were
not significant (data not shown).
Effectivel y, f
farther
quantifies the degree to which ovules
cause pollen tubes to deviate from random growth once
they come within a certain distance of a micropyle, but
this statistic does not address whether growth while
approaching this region is directed. To further analyze
pollen tube approach, we defined two angles: θ
mp
and
θ
tip
. The angle θ
tip
is the angle that a pollen tube turns as
it grows, and the angle θ
mp
is the angle that a pollen tube
would have to turn to grow directly toward the micropyle
(Fig. 2C). The difference between these two angles, Δθ =
θ
mp
- θ
tip
, measures how much pollen t ube growth devi-
ates from the most direct path toward the micro pyle (Δθ
= 0°). Owing to the periodic nature of angles, the distri-

nificantly different (p < 0.01) from the behavior of
pollen tubes approaching heat-treated ovules (Fig. 2D).
In each experiment, the pollen tubes grew similar dis-
tances before reaching the ovules, which indicates that
the difference in response results from the ovule incuba-
tion time. These data support a model where ovules
releaseadiffusiblesignal(attractant) throughout the
experiment, independently of the presence of pollen
tubes. The data also suggest a putative range over which
the resp onse operates: both the frequency f
closer
and the
distribution of the angle Δθ shows that pollen tubes that
grow within 50-100 μm of the micropyle of an ovule
show an increased reorientation to that ovule.
Furthermore, within 0-50 μm, pollen tubes appear to be
more directly guided to ovules with longer incubation
times. Although the operative range of attraction in
vitro mayvarywithdifferentexperimental conditions,
this range of 100 μm is larger than t he value of 33 ± 20
(s.d.) μm, which was based on observing when tubes
made sharp turns toward the micropyle under similar
agarose preparations [7].
The pollen tube response is consistent with following a
gradient
Previous studies have focused their analysis on only the
sharp, obvious turns that pollen tubes make near the
micropyle, both in vivo [22] and in vitro [7]. Here we
defineaquantitativemetric(theturningresponse)that
assesses t he mean turning behavior of the pollen tubes

(Fig.
3A), where G
tip
is in units of the change of concentra-
tion per unit distance. If a pollen tube is following a gra-
dient of attractant, then its turns should be correlated
with Δc/ΔL, and thus sin θ
mp
.
To test this hypo thesis, we looked at the relation
between θ
tip
and sinθ
mp
by fitting the line θ
tip
= A sin
θ
mp
+ ε (Fig. 3B) for the turns pollen tubes made at dif-
ferent distances from the micropyles of ovules that had
been incubated for different times (Fig. 3C). In each
case, there was a significant relation, as measured by the
Pearson r values and the slopes of the regression lines
(Table 2), at 50-100 and 100-150 μmfromthemicro-
pyle of ovules incubated for 0, 2, and 4 hours. At dis-
tances of 150-200 μm, there were still significant
correlations ( p < 0.05) for ovules incubated for 2 and 4
hours. As expected, datasets for the heat-treated ovules
did not show significant correlations. In all cases, the

). The data evidence two trends for this response:
it increases with longer incubation times and decreases
at farther distances from the micropyle (Fig. 3C).
Although pollen tubes are known to turn in response
to changes in their internal tip-focused cyto plasmic cal-
cium gradient [31], and gradients of small molecules
(ions and reactive species) affect pollen tube polarity
Figure 3 Pollen tube behavior is consistent with turning in
response to a gradient of an attractant across the tip surface.
(A) Schematic of gradient-following model. The pollen tube tip is
treated as flat. A gradient in the attractant (G
tip
) concentration gives
a difference in concentration Δc between the two sides of the tip.
(B) Fit of θ
tip
= Asin θ
mp
+ ε for points 0-50 μm from the micropyle
and 4-hour incubation time. In all fits, ε was not significantly
different from zero. The slope A can be considered the average
response of the pollen tubes to the ovule. (C) Fits were obtained at
varying distances from the closest micropyle: 0-50 μm, 50-100 μm,
100-150 μm, and 150-200 μm. The turning response (the slope A)
measures the average tendency for pollen tubes to turn toward the
micropyle based on the hypothesis that the turns sense a change in
the concentration of an attractant across the tip. Turning responses
are given for data collected with 0-, 2-, and 4-hour ovule incubation
times and also for heat-treated (boiled) ovules. Error bars are the
standard errors determined by the linear regression.

treated
0-50 -0.071 0.036 -0.091 54.97
50-100 -0.064 0.017 -0.08 32.52
100-150 0.051 0.015 0.112 9.16
150-200 -5.7 × 10
-
3
0.015 0.027 68.39
Responses reported are the unitless slope A of the regression line between
θ
tip
and sin θ
mp
. The column ΔResponse is the standard error of this slope.
The significance levels reported are for the Pearson r values: p <5%(•), p <
1% (••), p <0.1%(•••), p < 0.01% (••••).
Stewman et al. BMC Plant Biology 2010, 10:32
/>Page 6 of 20
and influence the direction of growth [8,31-34], the
mechanisms that couple external guidance cues to these
intracellu lar ion gradients remain unknown. Both spatial
and temporal sensing mechanisms have been suggested
in the litera ture on pollen tube guidance [1]. Our analy-
sis supports a spatial mechanism in which the pollen
tubes effectively measure t he concentration of the
attractant across their tips and turn accordingly. In the
temporal sensi ng that is characteristic of E. coli chemo-
taxis, a bacterium displays a series of runs that are sepa-
rated by isotropic tumbles [35-37]. This mechanism is
inconsistent with our findings, and would be hard to

interpretation of the experimental results.
We modeled how pollen tubes change their direction of
growth by splitting each turn into a directed and a ran-
dom component (Fig. 4A), which we assumed were inde-
pendent. The directed component specifies the mean
angle that a theoretical pollen tube would turn in
response to a gradient of the attractant, and the random
component adds a random angle chosen from a Gaussian
distribution to this mean direction. To determine the
directed component, w e assume that each bound recep-
tor ind uces a signal that gives the pollen tube some
propensity to turn in the direction of the receptor. For a
pollen tube to perceive a difference in the concentration
across its tip, there must be at least two patches of recep-
tors that are spatially separated on the pollen tube. Simi-
lar simple considerations have led to several successful
models of leukocyte chemotaxis (for example, [20,21]).
An exact model of spatial sensing would depend on both
the distribution of receptors in these patches (or across the
whole tip), the kinetics of the receptor-ligand interaction,
and the nature of the intracellular response that ultimately
results in the pollen tube turning. The distance and time
scales in our experiment are large enough that we can
assume receptors operate close to steady-state. We sim-
plify the remaining consideratio ns by assuming that the
change in c oncentration across the tip (Δc)ismuchless
than the average concentration at the tip (c), in which case
both the concentration along the tip and the difference in
bound receptors are approximately linear. The directed
component can then be approximated as proportional to

pollen tubes grow closer to the micropyle. The increase
in the gradient at longer incubation times implies
ongoing release at the source [45-47]. To simplify the
description of diffusion on the medium, we considered
only two-dimensional diffusion through the thin fluid
film that coats the surface of the medium and not
through the agar matrix itself.
Modeling the difference in concentration across the
tip of the pollen tube requires relating how the
Stewman et al. BMC Plant Biology 2010, 10:32
/>Page 7 of 20
concentration at the tip changes as the position of the
tip changes. As discussed in Section 2.2, we expect Δc/
ΔL = G
tip
sin θ
mp
(Fig. 3A). Consistent with our experi-
mental observations, G
tip
decreases with distance (the
turning response increases closer to the micropyle) and
increases with time (the turning response increases with
longer incubation times).
When we combine the mode l for the direct ion of pol-
len tube growth and the attractant gradient, there are
four parameters that describe the mean direction that
the tubes turn in response to an attractant: the turning
constant (), the rate of attractant production (k
p

approximately the same as that of ubiquitin, (8-9 kD),
which has a diffusion constant of 14.9 × 10
-7
cm
2
/sec
in aqueous solution [48]. Comparing the values is
complicated by the high sucrose content of the thin
film on top of the medium (18% w/v) and the possibi-
lityofnon-specificinteractions between the attractant
and the supporting agar. Both of these factors would
Figure 4 Model of pollen tube growth. (A) Conceptual depiction o f the directed and random components of turning. The directed
component (black arrow) is calculated based on the gradient of the attractant. The random component is a random angle added to this. The
gray shaded regions depict one standard deviation of the Gaussian distribution for the random angle. (B) Dynamics of a model of the gradient.
The model gives a theoretical concentration of the attractant (Eq. 3 in Methods), and the gradient is derived from this concentration. Here the
magnitude of the gradient from a single ovule, oriented toward the ovule micropyle is shown. Top: Depiction of the model for the attractant
gradient as a function of distance from the micropyle. The different curves (top to bottom) are for the gradient after the source has released the
attractant for 4.5 hours, 2.5 hours, and 0.5 hours. Bottom: Depiction of the model for the attractant gradient as a function of time on the
medium. The different curves (top to bottom) are for distances of 0 μm, 50 μm , 100 μm , and 150 μm from the micropyle.
Table 3 Parameters for the turning model
Parameter Description (units) Fit value 90% CI
’ Proportional response
(rad/conc min)
40.11 34.50-63.91
D Diffusion constant (μm
2
/min) 66.72 63.63-96.69
r
0
Radial offset (μm) 117.56 116.01-174.61

these two points is mathematically equivalent to 〈cos
θ
tip
(s, s + δs)〉. Plotting this quantity as a fun ction of δs
shows that it is approximately linear, and regression
yields an estimate for the persistence length of L =
1042.70 μm. The long persistence length indicates that
the probability of making a turn θ
tip
peaks sharply
around θ
tip
= 0, such that 〈cos θ
tip
〉 ≈ 1-〈θ
tip
2
〉 and that
the probability distribution can be described as a shar-
ply-peaked Gaussian with variance 〈θ
tip
2
〉 =2(δs/L)(see
Methods). The standard deviations predicted by this
form compared well with the circular standard devia-
tions of the actual angle distributions for Δt =20to
Δt = 100 min (Fig. 5C).
Figure 5 Validating the model. (A) Compar is on of exp erimental results with the model. T he responses are defined as in Fig. 3, where t he
response is the slope of the regression line between the turning angle θ
tip

way that we analyzed our experimental data (see Meth-
ods). We found that the mean responses (directed com-
ponent) in the simulations, as measured by the slope of
the regression line between θ
tip
and sin θ
mp
, compared
well to the data at different distances and for different
incubation times (Fig. 5A). We also assessed whether
the random growth seen in our simulations was com-
parable to that i n the experimen ts by analyzing the re si-
duals, differences betw een the θ
tip
predicted by the
regression and the actual θ
tip
.Wecomparedthestan-
dard deviations of the popula tions of these residuals for
both the simulations and the experiments (Table 4).
The standard deviations showed good agreement at dis-
tances far from the micr opyle (150-200 μm), where the
effects of an ovule should be small, and al so matched at
closer distances (100-150 μm) where there wa s a me a-
surableresponsetotheovules.Atevencloserdistances
(50-100 μm), the standard deviations compared wel l for
2-hour incu bation times and reasonably well for 4-hour
incubation times, but the experimental data had larger
standard deviations at 0-hour incubation times than did
our simulations. At the closest distances (0-50 μm), the

intervals for distances of 10-200 μm, and points within
5 μm of the interval center were included in the aver age
to reduce noise and help visualize the resulting trends.
We found that when pollen tubes approached functional
ovules, their rate of growth substantially decreased. This
decrease was not present when pollen tubes approac hed
heat-treated ovules, and the in cubation time of the
ovules influenced this decrea se by increasing the dis-
tance at which this slowing began (Fig. 6A). Specifically,
within 50 μm of the micropyle of heat-treated ovules,
poll en tubes grew at a rate of 2.29 ± 0.08 μm/min ; this
rate of growth decreased with the incubation time of
functional ovules, to 1.67 ± 0.11 μm/min around ovules
incubated for 4 hours (p < 0.001). Pollen tubes that
approach ovules with 0-hours of incubation did not
show a decrease in growth until very close to the micro-
pyle, while the decrease was apparent at a larger
distance for ovules with 2- and 4-hour incubation times.
The slowing partially explains the difference in observed
f
farther
frequencies at 0-50 μm.
In simulations, reducing the rate of growth increased the
ability of pollen tubes to target ovules
To explore how this reduced growth rate would influ-
ence the guidance process, we added terms to our simu-
lation to decrease the rate of growth with an increase in
Table 4 Comparison of variations in responses in
experiments and simulations.
Distance

calculated the fraction of tubes that we re successfully
able to target ovules in simulations for tubes t hat
included or excluded these terms (Fig. 6B). About 6-8%
of pollen tubes with no t urning or slowing (T-S-) were
still able to target the micropyle randomly, and when
slowing was enabled (T-S+), this frequency did not
change signific antly. When turning was enabled with no
slowing (T+S-), the frequency of tubes that would suc-
cessfully target more than doubled (from 6% to 20%
with a 4-hour incubat ion time), and this frequency
increased to over 60% when both were enabled (T+S+).
The difference between T+S- and T+S+ was visually
striking, in that tubes that reduced their rate of growth
showed substantially more guidance to the micropyle
than tubes that had a constant rate of growth (Fig. 6C).
Because the size of th e random turns in our model var-
ies with the growt h rate, we also simul ated pollen tubes
whose rate of g row th decreased with larger gradients of
the attractant, but whose random turns stayed the same
size (

tip
2
was initially calculated for a growth rate of
2.76 μm/min, but was kept constant). In these
simulations (T+S+ (*)), the fraction of tubes that were
successfully able to target ovules was close (differing by
less than 5%) to those where the random deviations
varied with the rate of growth (Fig. 6C), indicating that
the greater guidance we observed in simulations where

> 0°). If the pollen tube does not turn, over short times
(δt), the angle θ
mp
increases by δθ
mp
=(v/r )sin(θ
mp
)δt,
where v isthegrowthrateandr is the distance from
the micropyle. Thus the tube must turn toward the
micropylebymorethanthisamounttodecreaseθ
mp
.
For a turning response of A, δθ
tip
= A sin(θ
mp
)δt ,such
that the total change in the angle θ
mp
is
  
mp mp
v
r
At





medium. In the model, pollen tubes make turns that, on
average, follow the gradient of this attractant, but they
deviate from this path consistent with the random
growth observed when pollen tubes are grown with no
ovules present. This model successfully captures both
the directed and random behavior of the pollen tubes
growing in vitro and reveals that slowing growth near
the micropyle can greatly aid fertilization.
Although the recent identification of pollen tube
attractants in Torenia [10] is a significant step toward
understanding the molecular mechanism of guidance,
much still remains unknown. In particular, little is
known of the molecular mechanisms within the pollen
tube that enable sensing and responding to this attrac-
tant. Recent wor k on axon guidance identified an opti-
mal means of integrating signals from multiple
receptors [53]; in this model and the experiments to
validate it, the response depends in a complex way on
both the concentrati on of the attractant and the steep-
ness of the its gradient. The authors suggest a number
of possible molecular mechanisms that could give rise
to this behavior, and these may also be relevant to pol-
len tubes. A more complex relation between the
response and the concentration and steepness of the
gradient could also explain why there is no indication of
receptor saturation in our analysis, but more precise
control of the attractant gradient would be required to
validate this hypothesis.
Our model of the turning response did not assume a
particular molecular mechanism for the sensing process.

pollen tubes secrete a cell wall as they grow and can
only change direction by apical extension at the tip.
Stewman et al. BMC Plant Biology 2010, 10:32
/>Page 12 of 20
This mechani sm of growth enforces an internal polarity
that renders pollen tubes unable to respond isotropically
to a gradient, but this natural polarity may decrease the
size required to efficiently sense gradients, perhaps by
time-averaging of the number of bound receptors.
Our results support a long-r anged chemotropic model
for pollen tube guidance where pollen tubes respond to
stable g radients maint ained by ovul es continuously
releasing an attractant. Similar mechanisms have been
propo sed in genetic studies of the guidance proces s, but
it is unknown how the attraction observed in vitro will
manifest in vivo. Recent ge netic studies have uncovered
mutants that suggest a two-stage model for guidance in
the ovary [22,57-60]. In this model, pollen tubes show a
short-range of attraction near th e micropyle (micropylar
guidance) that is distinct from the longer-range gui-
dance that attracts them to grow along the funiculus
(funicular guidance) [22].
The range of attraction in vitro (100-150 μm) is
roughly the same as the range of attraction required of
funicular guidance in vivo. However, differences between
the in vitro and in vivo environments make these dis-
tances hard to compare. Our results strongly suggest
that pollen tubes follow a gradient of the attractant, in
which case an understanding of the factors that affect
this gradient is essential for relating the in vitro results

efficien t search strategies in man y organisms and would
explain the variance in pollen tube growth seen in the
ovary chamber [49,50]. Our measured persistence length
(~1 mm) is qualitatively consistent with in vivo ob serva-
tions that show pollen tubes grow essentially straight for
very long distances [49]. The distance over which
motion remains corr elated is unus ually long compared
with other cellular systems and comparable to the
length of the ovary chamber (2-3 mm long, see [61]).
Within the ovary, pollen tubes compete to find an
ovule; a long persistence of direction may provide an
efficient m ethod to locate a n unfertilized ovule for pol-
len tubes that emerge at the top of the chamber. It
would be interesting to determine if a correlation
existed between ovary length and pollen tube persis-
tence length in other species of the Brassicaceae family.
While it is unclear how t he decrease in the rate of
growth we observe near the micropyle in vitro manifests
in vivo, the dependence of this decrease on the incuba-
tion time suggests that the attractant mediates this
decrease, and simulations of our model suggest that it is
an important feature of guidance. The decrease in t he
growth rate could be responsible for the sharp turns
observed near the micropyle in vivo [22], and the
increased turning we observed in our simulations sug-
geststhatthisisaviablehypothesis.Thisresultdoes
suggest a potentially observable mutant phenotype: elim-
ination of the ability of pollen tubes to decrease their
rate of grow th would put them at a competitive disad-
vantage relative to wild-type.

removed from the ovary and deposited on the surface of
the medium, where they were then arranged around the
cut stigmas. A 001 insect pin mounted in a pin vice
(Fine Science Tools) was used for removal and subse-
quent manipulation of the ovules. The timing of polli-
nating the stigmas and placing the ovules was varied
according to the desired ovule incubation time
(Table 1). For 0-hour incubation times, the stigma was
pollinated and two hours later the ovules were placed.
For 2-hour i ncubation times, the ovules were placed,
and the stigmas were then immediately pollinated. For
4-hour incubation times, the stigmas were pollinated
two hours after the ovules were placed.
Microscopy
Time-lapse images of GFP-labeled pollen were acquired
using an Olympus Fluoview 1000 scanning confocal
microscope. Positions of the ovules and stigma were
determined using autofluorescence observed with a
Cy5.5 filter.
Correcting stack alignment
The total fluorescence measured at 540 nm in each
optical section was used to detect the surface of the
medium. Each slice Z in the stack had a measured total
fluorescence I(Z), and we normalized this with the maxi-
mum fluorescence in the stack

IZ IZ I() ()/
ma x

.The

(ΔZ).
Analysis of images
Polle n tube trajectori es were constructed by using Ima-
geJ image analysis software />download.html. Kalman filtering, as implemented by the
Kalman Stack Filter plugin to ImageJ by Chris Mauer,
was applied to the stacks before image analysis http://
rsb.info.nih.gov/ij/plugins/kalman.html. The tips of the
poll en tubes were identified manually. The micropyle of
each functional ovule was located by the point where a
pollen tube had entered the micropyle, penetrating the
ovule. This penetration was assessed with two
conditions: pollen tubes had to both reach the micro-
pyle, and subsequent growth had to occur within the
focal planes of the ovule autofluorescence. The micro-
pyles of heat-treated ovules were taken to be at the loca-
tion of the cleft where the funiculus joins the outer
integument of the ovule.
Except for the f
closer
and f
farther
frequencies, we only
included data from pollen tubes g rowing toward an
ovule that was eventually, but not yet, penetrated by a
pollen tube to ensure that our conclusions were based
on data for guidance toward functional, unfertilized
ovules. This restriction was not possible for t he heat-
treated control because the ovules were never pene-
trated in that case. The heat-treated control, in these
cases, allowed a comparison of growth of pollen tubes

mp
and θ
tip
were calculated using three posi-
tions (at t - Δt, t,andt + Δt) as follows. We calculated
the vector of the current growth direction: v
cur
(t)=r(t)-
r(t - Δt). The new growth direction was the calculated
similarly: v
new
( t)=r(t + Δt)-r(t). The direction to the
micropyle v
ov
(t) was calculated as in Δθ. For each point,
the angle between v
cur
(t) and v
new
(t) was denoted θ
tip
,
and the angle between v
cur
(t) and v
ov
(t) was denoted θ
mp
.
Descriptive statistics of angular data















11
N
x
N
y
i
i
i
i
cos sin .


The mean direction 〈Δθ〉 is apparent when 〈u〉 is
expressed in polar coordinates:
u 


cle. To see this, consider N an gles chosen from a
distribution with a very narrow spread around Δθ =0,
and M angles chosen from a distribution that is uniform
around the circle. In the narrow distribution, the unit
vectors u
i
will be almost identical, their sum will be a
vector with length close to N,andthelengthofthe
mean venctor 〈u〉 will be close to R ≈ 1, so s
0
≈ 0. In
the uniform distribution case, the vectors will be uni-
formly scattered so their directions will essentially can-
cle; the mean vector will be 〈u〉 ≈ 0, with length R ≈ 0,
and s
0
will diverge (s
0
® ∞).
Standard errors, confidence intervals, and tests for
statistical significance
Standard errors for f
closer
and f
farther
frequencies were
calculated by treating each as an estimate of a Bernoulli
trial probability, the standard error of which is
ffN()/1 
[63]. Significant differences between these

0
[1]
and s
0
[2] were the resampled circular standard devia-
tions, with 10,000 differ ent permutations to t est for sig-
nificantly different circular standard deviations.
Bootstrap calculations and permutatio n tests were per-
formed in the R analysis package [64-67]. The confi-
dence interval in the linear model describing persistence
was calculated from 10,000 samples generated using the
Monte Carlo method included in the program pro Fit
[68].
Standard errors in the θ
tip
angles were calculated by pro-
pagating the error in measuring the positions of each tip of
the tube. Using the standard propagation of uncertainty,
the standard error in the angle θ between two lines of
length l
1
and l
2
is given by: (s
θ
)
2
=(s
1
/l

tip














()
().
1
2
The first term describes turning that is proportional to
the difference in the fraction of receptors at steady-state
bound by the attractan t at ea ch side of the tip (Fig. 3A),
and the second term adds a random variation to the
receptor-mediated respo nse. Here  is the proportional-
ity constant for turning, c is the concentration of at trac-
tant in units of units of K
D
(C = cK
D
), Δc is the change

Grt
k
p
Dr r
tip
(,)
()
(,)
(











4
0
2
42
0
1

and
))
exp

where a finite amount o f attractant is deposited into an
infinitesimal region, making the concentration there infi-
nite [47].
To determine how experimental errors would affect
the model parameters, we used our Gaussian model for
the error in tip positions (s. d. of 4 μm) to generated
10,000 synthetic data sets. To estimate confidence inter-
vals for the model para meters, we fit these s data s ets.
Table 3 reports the 90% range of values for these fits.
Model of random turning
To access short regions of growth, we took advantage
of the fact that, on the length scales of the experiment,
pollentubegrowthissmoothwithfewsharpangles
and fit the points in the time-lapse with a spline curve
to study the growth at intervals as short as 5 μmof
growth. The change in direction of a pollen tube was
found by finding the angle between the direction of
growth at some distance along the tube s and a new
direction of growth a fter the tube had grown a dis-
tance δ s (Fig. 5B). We write this as the turning angle
θ
tip
(s, s + δs). Its cosine, cos θ
tip
(s, s + δs), measures
the correlation between the vector for the direction of
growth at s and the vector at s + δs: v(s)·v (s + δs)=
cosθ
tip
(s, s + δs), where v(s) has be en normalized to

-4
and L = 1074.02 μm. The
parameter b is the deviation from the intercept at unity
and was not statistically different from b =0.Thepara-
meter L is the persistence length, and was found to fall
in range 1042.70-1108.68 μm with 99.9% confidence.
Although the data showed non-random osci llations
around the linear fit, the deviations were small (the two
largest deviations were a 5% error at δs =115μmand
6% error at δs = 385 μm). The long persistence length L
indicates that the probability of making a turn θ
tip
peaks
sharply around θ
tip
= 0, indicating that 〈cosθ
tip
〉 ≈ 1-
〈θ
tip
2
〉. and that the distribution of θ
tip
can be described
as a sharply-peaked Gussian with mean 0 and variance
 
tip
ssL
2
2() ( /)








itip
tip
i
.
Here s
tip
includes both the error in measuring θ
tip
and
the fluctuations p redicted by the s parameter o f the model.
The term θ
i
is the predicted mean angle f or turning:
Stewman et al. BMC Plant Biology 2010, 10:32
/>Page 16 of 20
 
iimpi
citt (, (),)r
where the subsc ript i denotes each separate direction
to the micropyle θ
mp
(i), which has position r
i

p
/D)andfitting’ and the diffusion constant D
as two independent parameters (D can be left as a sepa-
rate parameter because of its appearance in the expo-
nential in Eq. 3). Written in terms of the parameters ’,
D, and r
0
the mean response θ
i
is then
 
imp
citDrt

(, (),; , )r
0
where we wri te
ctDrctkD
mp mp p
(, ,; , ) (, ,)/( / )rr

0

to emphasize that we removed the k
p
/D prefact or, but
that these terms are still parameterized by D and r
0
.To
understand the resulting fits, we assessed the turning

approaches v
min
. This formulation is consistent with our
general observation that once the rate of growth of a
pollen tube had slowed, it never substantially increased.
Our model is essentially a continuously-sampled formu-
lation of v
new
:
vt v v v
k
v
ct
t
() [( ) ]
()
min min
/








0
1
1


the model (Eqs. 1 and 4) by choosing discrete time-
steps of length Δt = 0.1 min and using the model equa-
tions to evolve the position and direction of growth of
each virtual pollen tube tip. Specifically, at each step in
time Δt, the simulation iterates through the list of pollen
tubes and does the following:
1. If the virtual pollen tube has been previously “cap-
tured” by coming within a short distance (10 μm)
of the virtual micropyle, or growing more than
800 μm from the center of the simulation, then
the simulation ignores it.
2. The virtual tip is advanced based on the previous
direction: r
j
(t + Δt)=r
j
(t)+v
j
(t)Δt
3. The concentration and gradient at position r
j
(t +
Δt)andtimet + Δt are calculated:
ct tt t
j
(( ), )r 
and
G ctttt
tip j
  (( ), )r 


sin ( , )v
,
where
  
j tip mp j tip j
GGz



sin ( , )v
is the gradi-
ent calculated in step 3 and θ
mp
is the angle
between the direction v
j
(t)ofthepreviousstep
and the gradient
  
j tip mp j tip j
GGz



sin ( , )v
.
5. In simulations where the rate of grow changes
(denoted S+ in the text), the n ew rate of growth
v



1
1

6. v
j
(t + Δt ) is generated by rotating v
j
(t)byδθ
j
(t +
Δt) and rescaling the vector to have magnitude v(t
+Δt).
7. If the new position of the virtual pollen tube, r
j
(t + Δt), is within 10 μm of the ovule icropyle or
has grown more than 800 μmfromthecenterof
the simulation, then the virtual tube is marked as
“ captured,” and is ignored in subsequent
iterations.
These steps are continuously iterated until the simula-
tion ends.
In addition to this algorithm, each simulation
requires the location of the virtual ovules and a set of
initial positions and directions for the virtual pollen
tubes. To enable direct comparison between our simu-
lations and the experimental data, we used the same
micropyle locations, initial pollen tube locations, and
incubation time as an experimental replicate. Each

would also like to thank Caroline Taylor, Maria Krisch, Beth Bray, and Mark
Johnson for their helpful comments on the manuscript.
Author details
1
Department of Chemistry, The University of Chicago, 929 E 57th St,
Chicago, IL 60637, USA.
2
James Franck Institute, The University of Chicago,
929 E 57th St, Chicago, IL 60637, USA.
3
Department of Molecular Genetics
and Cell Biology, CLSC 1106, 920 E 58th St, Chicago, IL 60637, USA.
4
Current
address: Chromatin, Inc, 3440 S Dearborn St, Suite 280, Chicago, IL 60616,
USA.
5
Institute for Biophysical Dynamics, The University of Chicago, 929 E
57th Street, Chicago, IL 60637, USA.
6
Department of Physics, The University
of Chicago, 5740 S Ellis Ave, Chicago, IL 60637, USA.
7
Department of Biology,
Knox College, 2 E South St, Galesburg, IL 61401-4999, USA.
8
International
Institute of Information Technology, Gachibowli, Hyderabad 500 032, Andhra
Pradesh, India.
9

/>Page 18 of 20
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